The Coercive Projection Law of Gravity: A Universal Variational Principle with Explicit Constants and Cross-Probe Falsifiers

We articulate a single, universal principle that governs gravitational inference under finite information: the coercive projection law. In this view, nature implements a projection from raw baryonic sources to an effective source through a fixed, scale- and time-aware kernel; the gravitational field is the unique minimizer of a classical energy with that effective source. We show that Information-Limited Gravity (ILG) is precisely the gravitational instantiation of this law in the pressure formulation, where the kernel w(k, a) = 1 + C (a/(kτ₀))^α maps observed baryons to the effective pressure p, and the potential Φ solves the classical Poisson equation ∇²Φ = 4πG a²p. Mathematically, we prove a coercivity inequality with explicit constants that (i) guarantees existence/uniqueness and stability of the projected solution, (ii) certifies positivity and monotonicity displays used in galaxies and cosmology, and (iii) yields falsifiers that bind probes together: rotation curves, tracer-independent EG, the low-ℓ ISW sign, and the low-L CMB-lensing amplitude are all simultaneously constrained by the same kernel and the same coercivity constants. Operationally, this converts "fits" into audited inferences: each analysis ships with machine-readable certificates (energy values, residual norms, positivity checks, convergence diagnostics) that verify compliance with the coercive projection law. Conceptually, the universality and the specific constants trace to Recognition Geometry: the exponent α = ½(1 − φ⁻¹) and prefactor C = φ⁻³/² are fixed by golden-ratio structure; the finite-net and rank-one projection constants match those that appear in independent CPM instantiations (e.g., protein folding), explaining cross-domain alignment. We release a minimal, dependency-light engine that implements the grid (FFT) and disk (Hankel) paths and emits certificates with each result (CPM-Cosmology-Grid-Path).